Refutation of Hilbert Space Fundamentalism

Quantum Mechanics (QM) represents the state of any closed system, which may even be the entire universe, as a vector $|\psi(t)\rangle$ in a complex Hilbert space $\mathcal{H}$. The Hamiltonian operator $\widehat{H}$ fully specifies its evolution equation,

where $\widehat{U}_{t,t_{0}}:=e^{-i/\hbar(t-t_{0})\widehat{H}}$ is the time evolution operator.

Since QM is invariant to unitary symmetries, this seems to justify the following:

The HSF Thesis is sometimes assumed more or less explicitly in some versions of various approaches to QM, like information-theoretic, decoherence, and Everett’s Interpretation. The sufficiency of the MQS is claimed, perhaps most explicitly, by Carroll and Singh [4], p. 95:

Everything else–including space and fields propagating on it–is emergent from these minimal elements.

But this goes beyond Everettianism. We read in [3]

The laws of physics are determined solely by the energy eigenspectrum of the Hamiltonian.

Scott Aaronson states in “The Zen Anti-Interpretation of Quantum Mechanics” [2] that a quantum state is

a unit vector of complex numbers […] which encodes everything there is to know about a physical system.

I confess that I think the HSF Thesis makes sense, if we take seriously the unitary symmetry of QM. Why would the position basis of the space $\mathcal{H}$ be fundamental, when it’s like other bases of $\mathcal{H}$? Just like the reference frames provide convenient descriptions of the $3$D-space with no physical reality, why would $|\mathbf{x}\rangle$ and $|\mathbf{p}\rangle$ be fundamental among all of the bases of $\mathcal{H}$? And if they play a privileged role, shouldn’t this role emerge from the MQS alone? Unfortunately, we will see that this cannot happen.

In [5] I gave a fully general proof that the HSF Thesis is not true: for any MQS, no emerging structure can be both unique and physically relevant. Here I will give a less abstract and less technical version of that proof.

Denote a preferred structure expected to uniquely emerge from the MQS by $\mathcal{S}_{\widehat{H}}^{|\psi\rangle}$, to express its dependence on $\widehat{H}$ and $|\psi\rangle$. Any such structure should be invariant, so it can be defined in terms of tensor objects from $\textstyle{\bigotimes}^{r}\mathcal{H}\otimes\textstyle{\bigotimes}^{s}\mathcal{H}^{\ast}$ [6].

For a structure to be considered “preferred”, it has to consist of tensors that satisfy specific conditions, expressed as invariant tensor equations or inequations. For example, a preferred basis can be given by a set of operators $\widehat{A}_{\alpha}\in\mathcal{H}\otimes\mathcal{H}^{\ast}$, satisfying the relations $\widehat{A}_{\alpha}\widehat{A}_{\alpha^{\prime}}-\widehat{A}_{\alpha}\delta_{\alpha\alpha^{\prime}}=0$, $\widehat{I}_{\mathcal{H}}-\sum_{\alpha}\widehat{A}_{\alpha}$=0, and $\operatorname{tr}\widehat{A}_{\alpha}=1$.

This is generally true, and justifies the following

In every case of interest, a set of Hermitian operators $\mathcal{S}_{\widehat{H}}^{|\psi\rangle}=\big{(}\widehat{A}_{\alpha}^{|\psi\rangle}\big{)}_{\alpha\in\mathcal{A}}$ suffices to specify the candidate preferred structure. We already saw this for a preferred basis. The $3$D-space can be given by the number operators for particles of each type $P$ at each point $\bm{x}$ in space, $\widehat{a}_{P}^{\dagger}(\bm{x})\widehat{a}_{P}(\bm{x})$. Tensor product structures of $\mathcal{H}$ can be specified in terms of local algebras of Hermitian operators. Therefore, we can limit to Hermitian operators only, avoiding much of the mathematical framework developed in [5] to include any kind of tensor objects.

Two state vectors $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ are physically equivalent, $|\psi_{1}\rangle\sim|\psi_{2}\rangle$ if they represent the same physical state, i.e. if they are related by gauge symmetry, space isometries etc. There is a group $G_{P}$ of unitary or anti-unitary transformations of $\mathcal{H}$, so that $|\psi_{1}\rangle\sim|\psi_{2}\rangle$ iff $|\psi_{2}\rangle=\widehat{g}|\psi_{1}\rangle$ for some $\widehat{g}\in G_{P}$. This equivalence extends uniquely to tensors on $\mathcal{H}$. The main claim of the HSF Thesis is:

Often the symmetry transformation from eq. (3) simply permutes the operators $\widehat{A}_{\alpha}^{|\psi\rangle}$. For example, space isometries permute the position operators.

To be physically relevant, a $3$D-space should be able to distinguish among physically distinct states, e.g., since the density $\langle\psi|\widehat{a}_{P}^{\dagger}(\bm{x})\widehat{a}_{P}(\bm{x})|\psi\rangle$ can be different for different values of $|\psi\rangle$, the same should be true for any candidate preferred $3$D-space. In particular, it should detect changes that the Hamiltonian cannot detect (the Hamiltonian cannot distinguish $|\psi\rangle$ from $|\psi^{\prime}\rangle$ iff there is a unitary $\widehat{S}$ so that $[\widehat{H},\widehat{S}]=0$ and $|\psi^{\prime}\rangle=\widehat{S}|\psi\rangle$, for example $\widehat{S}=\widehat{U}_{t_{1},t_{0}}$, but space should be able to distinguish them). The same applies to the components of $|\psi\rangle$ in a preferred basis, and to tensor factors of $\mathcal{H}$. Otherwise, such structures would have no physical relevance.

Physical equivalence requires that the structure has to distinguish not merely the unit vectors $|\psi\rangle\nsim|\psi^{\prime}\rangle\in\mathcal{H}$, but any other state vectors $\widehat{g}|\psi\rangle$ and $\widehat{g^{\prime}}|\psi^{\prime}\rangle\in\mathcal{H}$ representing the same physical states, for any $\widehat{g},\widehat{g^{\prime}}\in G_{P}$.

I now prove that these two conditions are incompatible.

Example 1 gives an infinite family of physically distinct ways to choose the $3$D-space. But in fact there is such an infinite family for each generator commuting with $\widehat{H}$.

Theorem 1 was applied explicitly to various constructions assumed by the HSF Thesis to emerge from the MQS, and showed that, if they are physically relevant, they are not unique [5]. In particular, uniqueness fails for the generalized preferred basis (including for subsystems), the tensor product structure, emergent $3$D-space, emergent classicality etc. Such structures cannot emerge uniquely from the MQS, a choice is always required.

A question stands: what breaks the symmetry of QM?